Let's work an example of a Cross Sum puzzle. To make it easier
to explain the process, the boxes have been labeled A through I. You may
also find this useful when you work the puzzles yourself.
Example
| | ÷ | | × | | 20 |
| × | | + | | × | |
| | × | | × | | 189 |
| - | | + | | - | |
| | ÷ | | + | | 10 |
| 20 | | 10 | | 39 | |
|
Labels
| A | ÷ | B | × | C | 20 |
| × | | + | | × | |
| D | × | E | × | F | 189 |
| - | | + | | - | |
| G | ÷ | H | + | I | 10 |
| 20 | | 10 | | 39 | |
|
On the top row we have A÷B×C equal to 20. So A÷B and C
are two numbers that multiply to make 20. The only numbers that can be
multiplied to make 20 are 1×20, 2×10, 4×5, 5×4,
10×2 and 20×1. Since the highest possible value for A÷B
is 9÷1, which is 9, we can eliminate 10×2 and 20×1. Since
the highest possible value for C is also 9 we can eliminate 1×20 and
2×10. So either A÷B is 4 and C is 5, or A÷B is 5
and C is 4.
There are only a few ways that A÷B could be 4 or 5, namely
4÷1, 8÷2, or 5÷1. This means that B can only be
1 or 2, and that C can only be 4 or 5. Let's put these all together,
and look at the possibilities.
| 4 | ÷ | 1 | × | 5 | 20 |
| × | | + | | × | |
| D | × | E | × | F | 189 |
| - | | + | | - | |
| G | ÷ | H | + | I | 10 |
| 20 | | 10 | | 39 | |
|
| 8 | ÷ | 2 | × | 5 | 20 |
| × | | + | | × | |
| D | × | E | × | F | 189 |
| - | | + | | - | |
| G | ÷ | H | + | I | 10 |
| 20 | | 10 | | 39 | |
|
| 5 | ÷ | 1 | × | 4 | 20 |
| × | | + | | × | |
| D | × | E | × | F | 189 |
| - | | + | | - | |
| G | ÷ | H | + | I | 10 |
| 20 | | 10 | | 39 | |
|
Next let's look at the right column. We have C×F-I equal
to 39. Here, C must be 4 or 5. We can immediately eliminate 4 because
in order for 4×F-I to be 39, F would have to be at least 10. This
is impossible since all of the numbers must be from 1 to 9. So C must
be 5. Now C×F-I must be either 5×8-1 or 5×9-6.
We can eliminate 5×8-1 because of the middle row, D×E×F
which equals 189. If F were 8, then D×E×F would be even, but
189 is an odd number, so F cannot be 8. This means C×F-I has to be
5×9-6. The possibilities are now
| 4 | ÷ | 1 | × | 5 | 20 |
| × | | + | | × | |
| D | × | E | × | 9 | 189 |
| - | | + | | - | |
| G | ÷ | H | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|
| 8 | ÷ | 2 | × | 5 | 20 |
| × | | + | | × | |
| D | × | E | × | 9 | 189 |
| - | | + | | - | |
| G | ÷ | H | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|
In each case there is only one possiblity for the left column A×D-G
to equal 20. These are 4×7-8 and 8×3-4. This gives us
| 4 | ÷ | 1 | × | 5 | 20 |
| × | | + | | × | |
| 7 | × | E | × | 9 | 189 |
| - | | + | | - | |
| 8 | ÷ | H | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|
| 8 | ÷ | 2 | × | 5 | 20 |
| × | | + | | × | |
| 3 | × | E | × | 9 | 189 |
| - | | + | | - | |
| 4 | ÷ | H | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|
Now there is only one blank space left in each row, and we can just
fill those in by sight to make the rows work out.
| 4 | ÷ | 1 | × | 5 | 20 |
| × | | + | | × | |
| 7 | × | 3 | × | 9 | 189 |
| - | | + | | - | |
| 8 | ÷ | 2 | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|
| 8 | ÷ | 2 | × | 5 | 20 |
| × | | + | | × | |
| 3 | × | 7 | × | 9 | 189 |
| - | | + | | - | |
| 4 | ÷ | 1 | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|
In the left grid above, the middle column is 1+3+2 which does not equal 10,
while the right grid has 2+7+1 which is 10. So the answer is
| 8 | ÷ | 2 | × | 5 | 20 |
| × | | + | | × | |
| 3 | × | 7 | × | 9 | 189 |
| - | | + | | - | |
| 4 | ÷ | 1 | + | 6 | 10 |
| 20 | | 10 | | 39 | |
|